Hi, When assessing the fit of a GMM, is it possible to test whether a slope or intercept variance should be free, or held equal to zero, using a chi-square difference test on the difference in the loglikelihoods?

I would not do a difference test in this way with a mixture model. I would use MODEL TEST to test whether coefficients are equal. I would not test if a variance is zero. This is on the border of the permissible parameter space.